Notes on mathematics, optimal transport, machine learning, and scientific life, by Gabriel Peyré. https://www.gpeyre.com/blog/ Sat, 03 Oct 2026 00:00:00 +0000 en Jekyll https://www.gpeyre.com/blog/2026/10/03/inverse-optimisation-gap-loss/ https://www.gpeyre.com/blog/2026/10/03/inverse-optimisation-gap-loss/ Sat, 03 Oct 2026 00:00:00 +0000 Gabriel Peyré en The convex optimality gap as a Fenchel–Young loss, with an interactive inverse optimal transport experiment between two Gaussian mixtures and a linked bibliography. Optimal transport https://www.gpeyre.com/blog/2026/09/15/gaussian-preserving-wasserstein-flows/ https://www.gpeyre.com/blog/2026/09/15/gaussian-preserving-wasserstein-flows/ Tue, 15 Sep 2026 00:00:00 +0000 Gabriel Peyré en Gelbrich's Gaussian contraction and a criterion explained by Hugo Lavenant, with proofs for Gaussian relative entropy and heat flow. Optimal transport https://www.gpeyre.com/blog/2026/09/15/ia-pour-les-maths-clarifications/ https://www.gpeyre.com/blog/2026/09/15/ia-pour-les-maths-clarifications/ Tue, 15 Sep 2026 00:00:00 +0000 Gabriel Peyré fr Une clarification de mon intervention sur France Inter, en français et en anglais. Science & société https://www.gpeyre.com/blog/2026/08/14/polyak-lojasiewicz-gradient-flows/ https://www.gpeyre.com/blog/2026/08/14/polyak-lojasiewicz-gradient-flows/ Fri, 14 Aug 2026 00:00:00 +0000 Gabriel Peyré en Why a gradient inequality can be enough for exponential convergence, illustrated by an interactive curved valley. Optimization https://www.gpeyre.com/blog/2026/08/07/mathematical-nexus/ https://www.gpeyre.com/blog/2026/08/07/mathematical-nexus/ Fri, 07 Aug 2026 00:00:00 +0000 Gabriel Peyré en A collection of mathematical animations and accompanying Python notebooks, now gathered in the Mathematical Nexus. Resources https://www.gpeyre.com/blog/2026/07/30/perron-frobenius-and-sinkhorn/ https://www.gpeyre.com/blog/2026/07/30/perron-frobenius-and-sinkhorn/ Thu, 30 Jul 2026 00:00:00 +0000 Gabriel Peyré en Compare a positive Markov operator with Sinkhorn’s nonlinear scaling map in an interactive simplex. Optimal transport https://www.gpeyre.com/blog/2026/07/13/kl-barycenters-of-gaussians/ https://www.gpeyre.com/blog/2026/07/13/kl-barycenters-of-gaussians/ Mon, 13 Jul 2026 00:00:00 +0000 Gabriel Peyré en A short derivation of the reverse-KL Gaussian barycenter, with an animated two-dimensional example. Probability https://www.gpeyre.com/blog/2026/07/10/discrete-duhamel-formula/ https://www.gpeyre.com/blog/2026/07/10/discrete-duhamel-formula/ Fri, 10 Jul 2026 00:00:00 +0000 Gabriel Peyré en Derive the discrete Duhamel identity and use it to propagate local perturbations through a matrix product. Linear algebra https://www.gpeyre.com/blog/2026/07/07/legendre-transforms-and-moment-measures/ https://www.gpeyre.com/blog/2026/07/07/legendre-transforms-and-moment-measures/ Tue, 07 Jul 2026 00:00:00 +0000 Gabriel Peyré en Why minus the log-integral of the exponential of a negative convex conjugate is convex, and how its variation produces a moment measure. Convexity https://www.gpeyre.com/blog/2026/07/04/doubly-nonnegative-completely-positive/ https://www.gpeyre.com/blog/2026/07/04/doubly-nonnegative-completely-positive/ Sat, 04 Jul 2026 00:00:00 +0000 Gabriel Peyré en A five-by-five matrix and an elementary proof of the gap between doubly nonnegative and completely positive matrices. Linear algebra https://www.gpeyre.com/blog/2026/06/16/optimal-transport-for-machine-learners/ https://www.gpeyre.com/blog/2026/06/16/optimal-transport-for-machine-learners/ Tue, 16 Jun 2026 00:00:00 +0000 Gabriel Peyré en The alpha version of my new book is online, including interactive figures. Books https://www.gpeyre.com/blog/2026/06/15/diffusion-versus-optimal-transport/ https://www.gpeyre.com/blog/2026/06/15/diffusion-versus-optimal-transport/ Mon, 15 Jun 2026 00:00:00 +0000 Gabriel Peyré en Compare the deterministic score-based probability flow with quadratic optimal transport from a centered Gaussian to a three-Gaussian mixture, down to the three-Dirac limit. Diffusion models https://www.gpeyre.com/blog/2026/05/06/covariance-gauges-and-matrix-norms/ https://www.gpeyre.com/blog/2026/05/06/covariance-gauges-and-matrix-norms/ Wed, 06 May 2026 00:00:00 +0000 Gabriel Peyré en A detailed proof that a positive convex homogeneous covariance gauge lifts to a matrix norm exactly when it is monotone. Matrix analysis https://www.gpeyre.com/blog/2026/04/16/straight-flows-and-euler-error/ https://www.gpeyre.com/blog/2026/04/16/straight-flows-and-euler-error/ Thu, 16 Apr 2026 00:00:00 +0000 Gabriel Peyré en A detailed Euler error bound isolating trajectory acceleration, with a proof and an interactive example. Differential equations https://www.gpeyre.com/blog/2026/04/06/muon-spectral-wasserstein/ https://www.gpeyre.com/blog/2026/04/06/muon-spectral-wasserstein/ Mon, 06 Apr 2026 00:00:00 +0000 Gabriel Peyré en How Muon's polar update connects to a Wasserstein gradient flow defined through displacement covariances. Optimal transport https://www.gpeyre.com/blog/2025/08/09/positive-definite-congruence/ https://www.gpeyre.com/blog/2025/08/09/positive-definite-congruence/ Sat, 09 Aug 2025 00:00:00 +0000 Gabriel Peyré en An explicit solution of UYU = X, with proofs of uniqueness and smooth dependence on positive definite matrices. Linear algebra https://www.gpeyre.com/blog/2025/07/02/linear-rank-preservers/ https://www.gpeyre.com/blog/2025/07/02/linear-rank-preservers/ Wed, 02 Jul 2025 00:00:00 +0000 Gabriel Peyré en Why every linear rank preserver is a left-right change of basis, possibly composed with transposition in the square case. Linear algebra https://www.gpeyre.com/blog/2025/03/28/tout-comprendre-intelligence-artificielle/ https://www.gpeyre.com/blog/2025/03/28/tout-comprendre-intelligence-artificielle/ Fri, 28 Mar 2025 00:00:00 +0000 Gabriel Peyré en A personal recommendation of Olivier Cappé and Claire Marc's illustrated book on artificial intelligence, forthcoming from CNRS Éditions. Books